Abstract:

We consider the problem of performing inference for a continuous treatment effect on a binary outcome variable while controlling for high dimensional baseline covariates. We propose a novel Bayesian framework for performing inference for the desired low dimensional parameter in a high dimensional logistic regression model. While it is relatively easier to address this problem in linear regression, the nonlinearity of the logistic regression poses additional challenges that make it difficult to orthogonalize the effect of the treatment variable from the nuisance variables. Our proposed approach provides the first Bayesian alternative to the recent frequentist developments and can incorporate available prior information on the parameters of interest, which plays a crucial role in practical applications. In addition, the proposed approach incorporates uncertainty in orthogonalization in high-dimensions instead of relying on a single instance of orthogonalization as done by frequentist methods. We provide uniform convergence results that show the validity of credible intervals resulting from the posterior. Our method has competitive empirical performance when compared with state-of-the-art methods.

Abstract:

In matched observational studies, the inferred causal conclusions pretending that matching has taken into account all confounding can be sensitive to unmeasured confounding. In such cases, a sensitivity analysis is often conducted, which investigates whether the observed association between treatment and outcome is due to effects caused by the treatment or it is due to hidden confounding. The Rosenbaum sensitivity analysis is a modern approach for conducting sensitivity analysis for matched observational studies. It investigates what magnitude the maximum of the hidden biases from all matched sets needs to be in order to explain away the observed association, assuming that the treatment has no effect.
However, such a sensitivity analysis can be overly conservative and pessimistic, especially when the investigators believe that some matched sets may have exceptionally large hidden biases. In this paper, we generalize Rosenbaum’s framework to conduct sensitivity analysis on quantiles of hidden biases from all matched sets, which are more robust than the maximum. Moreover, we demonstrate that the proposed sensitivity analysis on all quantiles of hidden biases is simultaneously valid and is thus a free lunch added to the conventional sensitivity analysis. Finally, we demonstrate that the proposed sensitivity analysis also works for bounded null hypotheses as long as the test statistic satisfies certain properties.